Time-frequency analysis on the adeles over the rationals

Enstad, Ulrik B.R.; Jakobsen, Mads S.; Luef, Franz

doi:10.1016/j.crma.2018.12.004

Harmonic analysis

Time-frequency analysis on the adeles over the rationals
[Analyse des temps-fréquences sur l'anneau des adèles des rationnels]

Présenté par : the Editorial Board

Enstad, Ulrik B.R.¹ ; Jakobsen, Mads S.² ; Luef, Franz ²

¹ University of Oslo, Department of Mathematics, Oslo, Norway
² Norwegian University of Science and Technology, Department of Mathematical Sciences, Trondheim, Norway

Comptes Rendus. Mathématique, Tome 357 (2019) no. 2, pp. 188-199.

Résumé
Abstract

Nous montrons que la construction de trames (ou repères) de Gabor de $L^{2} (R)$ avec générateurs dans $S_{0} (R)$ et des décalages de temps-fréquence dans un réseau rectangulaire $α Z \times β Z$ est équivalente à la construction de certaines trames de Gabor pour $L^{2}$ sur les adèles des rationnels avec le groupe $R \times Q_{p}$ . Nous analysons également les relations entre la construction de trames de Gabor sur les adèles et sur $R \times Q_{p}$ et la construction de certains modules de Heisenberg.

We show that the construction of Gabor frames in $L^{2} (R)$ with generators in $S_{0} (R)$ and with respect to time-frequency shifts from a rectangular lattice $α Z \times β Z$ is equivalent to the construction of certain Gabor frames for $L^{2}$ over the adeles over the rationals and the group $R \times Q_{p}$ . Furthermore, we detail the connection between the construction of Gabor frames on the adeles and on $R \times Q_{p}$ with the construction of certain Heisenberg modules.

Reçu le : 2018-07-27
Accepté le : 2019-01-07
Publié le : 2019-01-28

DOI : 10.1016/j.crma.2018.12.004

Affiliations des auteurs :

Enstad, Ulrik B.R. ¹ ; Jakobsen, Mads S. ² ; Luef, Franz ²

¹ University of Oslo, Department of Mathematics, Oslo, Norway
² Norwegian University of Science and Technology, Department of Mathematical Sciences, Trondheim, Norway

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     title = {Time-frequency analysis on the adeles over the rationals},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {188--199},
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Enstad, Ulrik B.R.; Jakobsen, Mads S.; Luef, Franz. Time-frequency analysis on the adeles over the rationals. Comptes Rendus. Mathématique, Tome 357 (2019) no. 2, pp. 188-199. doi : 10.1016/j.crma.2018.12.004. http://www.numdam.org/articles/10.1016/j.crma.2018.12.004/

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